Stabilization of the population of a parasite on fruit by the population of a cleptoparasite

Abstract

When there exist two species such that one is a parasite on fruit and the other exploits the parasitized fruits, they must compete for a limited resource with each other. The relation between Dacus cucurbitae and Atherigona orientalis is an example of such a situation. We raise a question whether the population of a parasite on fruit can be stabilized by the existence of the cleptoparasite of the parasite on fruit. The changes in their population densities are represented as a differential equation with time delayed parameters, which is deduced from the context of life histories of the two species. An index representing degree of overlapping of generations (g) is defined as an average oviposition period devided by the average preoviposition period, and the value is assumed to be the same in the two species. The stability of the system is classified by three parameters: the reproductive rate of the parasite on fruits (R), the survival probability of it through competition against the cleptoparasite (p), and the generation overlapping index (g). For small values of g, e.g. less than some 0.5, the stability is determined mainly by a product of Rp: the values larger than 1 result in no equilibrium and infinite increase of both species, the values near 0 lead to large amplitude oscillations, and the intermediate values make stable equilibria or regular small oscillations. As g takes the larger values, the stability region in the space (p, R) occupies the larger area. The model presented here is well adjusted to the fluctuating pattern of the population of D. cucurbitae on Okinawa Is., and would also be applied to analysis of both hyperparasitisms and inquilin.